Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511802 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×B(4,𝔽2) |
| commutator, Frattini, cosocle | C22×Q8 | C24 |
| prime radical | #23910 | C2 |
| LCS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
1066 conjugacy classes of subgroups; only the 140 normal subgroups are shown.
This group has 1066 conjugacy classes of subgroups — too many to draw. Only its 140 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,16)(13,17)(14,20)(18,24)(19,26)(21,31)(22,32)(23,29)(25,33)(27,34)(28,35)(30,36)(37,40)(38,39);(2,5)(6,8)(9,11)(10,15)(12,22)(13,23)(14,21)(16,32)(17,29)(18,28)(19,30)(20,31)(24,35)(25,37)(26,36)(27,38)(33,40)(34,39);(1,3,4,7)(2,6,5,8)(10,16)(11,21)(12,24)(13,26)(15,28)(17,33)(18,20)(19,34)(29,39)(30,40)(31,32)(37,38);(10,17)(13,26)(15,29)(16,33)(18,34)(19,20)(23,36)(25,27)(28,39)(30,31)(32,40)(37,38)Polycyclic presentation:
2^9|c^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi,g^a=gi,h^a=hi,c^b=ci,f^c=fi,g^c=ghi,f^d=fh,g^d=gi,g^f=gi53 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419328);